Space of modular forms of level 45, weight 10, and trivial character

• Label: 45.10.a
• Level: $45$
• Weight: $10$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $8$
• Sturm bound: $60$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(60\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	58	15	43
Cusp forms	50	15	35
Eisenstein series	8	0	8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(4\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(8\)

Trace form

\( 15 q - 50 q^{2} + 4352 q^{4} - 625 q^{5} + 2294 q^{7} - 55332 q^{8} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(45))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
45.10.a.a	$45$	$10$	45.a	1.a	$1$	$1$	$1$	$23.177$	\(\Q\)	None	✓				15.10.a.b	$1$	$1$	\(-22\)	\(0\)	\(625\)	\(-5988\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-22q^{2}-28q^{4}+5^{4}q^{5}-5988q^{7}+\cdots\)
45.10.a.b	$45$	$10$	45.a	1.a	$1$	$1$	$1$	$23.177$	\(\Q\)	None	✓				15.10.a.a	$1$	$0$	\(4\)	\(0\)	\(-625\)	\(-7680\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-496q^{4}-5^{4}q^{5}-7680q^{7}+\cdots\)
45.10.a.c	$45$	$10$	45.a	1.a	$1$	$1$	$1$	$23.177$	\(\Q\)	None	✓				5.10.a.a	$1$	$1$	\(8\)	\(0\)	\(625\)	\(4242\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-448q^{4}+5^{4}q^{5}+4242q^{7}+\cdots\)
45.10.a.d	$45$	$10$	45.a	1.a	$1$	$2$	$2$	$23.177$	\(\Q(\sqrt{241}) \)	None	✓				15.10.a.d	$1$	$0$	\(-31\)	\(0\)	\(-1250\)	\(14112\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-2^{4}-\beta )q^{2}+(286+31\beta )q^{4}-5^{4}q^{5}+\cdots\)
45.10.a.e	$45$	$10$	45.a	1.a	$1$	$2$	$2$	$23.177$	\(\Q(\sqrt{4729}) \)	None	✓		✓		15.10.a.c	$1$	$1$	\(-19\)	\(0\)	\(1250\)	\(-11872\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-9-\beta )q^{2}+(751+19\beta )q^{4}+5^{4}q^{5}+\cdots\)
45.10.a.f	$45$	$10$	45.a	1.a	$1$	$2$	$2$	$23.177$	\(\Q(\sqrt{1009}) \)	None	✓				5.10.a.b	$1$	$0$	\(10\)	\(0\)	\(-1250\)	\(1700\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{2}+(522-10\beta )q^{4}-5^{4}q^{5}+\cdots\)
45.10.a.g	$45$	$10$	45.a	1.a	$1$	$3$	$3$	$23.177$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	45.10.a.g	$1$	$1$	\(-25\)	\(0\)	\(-1875\)	\(3890\)		$+$	$+$	$+$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-8+\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots\)
45.10.a.h	$45$	$10$	45.a	1.a	$1$	$3$	$3$	$23.177$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	45.10.a.g	$1$	$0$	\(25\)	\(0\)	\(1875\)	\(3890\)		$+$	$-$	$-$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(8-\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(45)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)